Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 364079, 26 pages
doi:10.1155/2010/364079
Research Article
Existence and Multiplicity of Solutions to Discrete
Conjugate Boundary Value Problems
Bo Zheng
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou,
Guangdong 510006, China
Correspondence should be addressed to Bo Zheng, zhengbo611@yahoo.com.cn
Received 21 March 2010; Accepted 29 April 2010
Academic Editor: Yong Zhou
Copyright q 2010 Bo Zheng. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We consider the existence and multiplicity of solutions to discrete conjugate boundary value
problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which
includes the asymptotically linear as a special case. By classifying the linear systems, we define
index functions and obtain some properties and the concrete computation formulae of index
functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by
combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory.
Our results are new even for the case of asymptotically linear.
1. Introduction
Let N, Z, and R be the sets of all natural numbers, integers, and real numbers, respectively.
For a, b ∈ Z, define Za, b {a, a 1, . . . , b} when a ≤ b. Δ is the forward difference operator
defined by Δun un 1 − un, and Δ2 un ΔΔun. Let A be an m × m matrix. Aτ or
xτ denotes the transpose of matrix A or vector x. The set of eigenvalues of matrix A will be
denoted by σA, and the determinant of matrix A will be denoted by det A.
Discrete boundary value problems BVPs for short arise in the study of solid state
physics, combinatorial analysis, chemical reactions, population dynamics, and so forth.
Besides, they are also natural consequences of the discretization of continuous BVPs. Thus,
these problems have been studied by many scholars.
Discrete two-point BVPs
Δ2 un − 1 fn, un 0,
u0 A,
n ∈ Z1, T ,
uT 1 B
1.1
2
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often appear in electrical circuit analysis, mathematical physics, finite elasticity, and so forth
as the mathematical models, where f : Z1, T × Rd → Rd with d ∈ N, T > 0 is a given integer,
and A, B are given constants.
We may think of 1.1 as being a discrete analogue of the continuous BVPs:
u ft, u 0,
ua A,
a ≤ t ≤ b,
ub B,
1.2
which have been studied by many scholars because of its numerous applications in science
and technology. In particular, Hale, Walter, Mawhin, and so forth have obtained some
significant results on the existence, uniqueness, and multiplicity of solutions of 1.2. We refer
the readers to 1–3 and references therein for further details.
Let
yn Bn − An − T − 1
,
T 1
zn un − yn.
1.3
Then 1.1 reduces to
Δ2 zn − 1 f n, zn yn 0,
n ∈ Z1, T ,
z0 0 zT 1.
1.4
Hence, in the following, we can only consider the discrete conjugate BVPs, that is,
Δ2 un − 1 fn, un 0,
n ∈ Z1, T ,
u0 0 uT 1.
1.5
As being remarked in 4, the nature of the solution of a continuous problem is not
identical with that of the solution of its discrete analogue. And since discrete analogs of
continuous problems yield interesting dynamical systems in their own right, many scholars
have investigated BVPs 1.5 independently. There are fundamental questions that arise for
BVPs 1.5. Does a solution exist, is it unique, and how many solutions can be found if
BVPs 1.5 have multiple solutions? How to find the lower bound or the upper bound of
the number of solutions of BVPs 1.5? Furthermore, how to obtain the precise number of
solutions of BVPs 1.5?
In recent years, the existence, uniqueness, and multiplicity of solutions of discrete
BVPs have been studied by many authors. In fact, early in 1968, Lasota 5 studied the
discretizations of 1.2 with ft, u replaced by ft, u, u and proved that the discrete problem
had one and only one solution with f satisfying a Lipschitz condition. Note that under certain
conditions the solution of a nonhomogeneous BVPs can be expressed in terms of Green’s
functions. For example, suppose that un is a solution of 1.1. Then
un T
Gn, mfm, um wn,
m1
1.6
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3
where Gn, m is Green’s function for
Δ2 un − 1 0,
n ∈ Z1, T ,
u0 0 uT 1,
wn A 1.7
B−A
n.
T 1
Let B {u | u is a real-valued function defined on Z0, T 1, u0 uT 1 0}, and define
T : B → B by
Tun T
Gn, mfm, um wn
1.8
m1
for n in Z0, T 1. Then there is a one-to-one correspondence between the fixed points of T
and the solutions of BVPs 1.1. When the nonlinearity f satisfies growth conditions known
as Lipschitz conditions, a unique solution of BVPs 1.1 can be obtained by using Contraction
Mapping Theorem see 6, 7 for more details.
Note that discrete BVPs model numerous physical phenomena in nature hence it is
of fundamental importance to know the criteria that ensure the existence of at least one
meaningful solution. And since discrete BVPs often have multiple solutions, it is useful to
have a collection of results that yield existence of solutions without the implication that the
solutions must be unique. To this end, many scholars have obtained some significant results
on the existence and multiplicity of solutions of discrete BVPs by using various analytic
techniques and various fixed-point theorems, for example, the upper and lower solution
method 8–10, the conical shell fixed point theorems 11, 12, the Brouwer and Schauder
fixed point theorems 9, 13, 14, and topological degree theory 15, 16. As we know, criticalpoint theory which includes the minimax method and Morse theory, etc. has played an
important role in dealing with the existence and multiplicity of solutions to continuous
systems 2, 17. It is natural for us to think that critical-point theory may be applied to study
the existence and multiplicity of solutions to discrete systems. In fact, in recent papers 18–
25, the authors have applied critical-point theory to study the existence and multiplicity of
periodic solutions to discrete systems. We also refer to 26–31 for the discrete BVPs. In 26,
Agarwal et al. employed the Mountain Pass Lemma to study 1.5 and obtained the existence
of multiple solutions. Very recently, B. Zheng and Q. Zhang 32 studied discrete BVPs 1.5
with fn, un V un and obtained the existence of exactly three solutions by using both
Morse theory and degree theory, and so forth. To the best of our knowledge, 32 is among
a few works dealing with discrete BVPs by using Morse theory. Hence, further studies on
application of Morse theory to discrete BVPs are still perspective.
Here, we consider the case fn, un ∇V n, un that is, we consider the following
discrete conjugate BVPs:
Δ2 un − 1 ∇V n, un 0,
u0 0 uT 1,
n ∈ Z1, T ,
1.9
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where V n, · ∈ C1 Rd , R for every n ∈ Z1, T , ∇V n, z denotes the gradient of V with
respect to z, and d ≥ 2, T > 0 are given integers.
Assume
∇V n, z An, zz o|z|
1.10
as |z| → ∞, where A : Z1, T × Rd → GLs Rd , and
A1 n ≤ An, z ≤ A2 n
1.11
for every n ∈ Z1, T , and z ∈ Rd , A1 , A2 : Z1, T → GLs Rd , GLs Rd denotes the group
of d × d real nonsingular symmetric matrices, and |z| denotes the Euclidean norm of z in Rd .
Throughout this paper, for any A1 , A2 ∈ GLs Rd , we denote A1 ≤ A2 if A2 − A1 is semipositive definite, and we denote A1 < A2 if A2 − A1 is positive definite. For any A1 , A2 :
Z1, T → GLs Rd , we denote A1 ≤ A2 if A1 n ≤ A2 n for every n ∈ Z1, T , and we
∅.
denote A1 < A2 if A1 n ≤ A2 n for every n ∈ Z1, T and {n | A1 n < A2 n} /
If An, z ≡ An in 1.10, then 1.10 is usually called an asymptotically linear
condition. So here we call 1.10 and 1.11 generalized asymptotically linear conditions. Our
results are new even for the case of asymptotically linear case.
The rest of this paper is organized as follows. In Section 2, firstly, we classify the linear
systems
Δ2 un − 1 Anun 0,
n ∈ Z1, T ,
u0 0 uT 1
1.12
for every A : Z1, T → GLs Rd . This classification gives a pair of integers iA, νA ∈
Z0, dT × Z0, d. We call iA and νA the index and nullity of A, respectively. Secondly,
we give some properties of the index and nullity together with the concrete computation
formulae. And finally, we introduce the definition of relative Morse index and give its precise
description. By using both results in Section 2 and Leray-Schauder principle, we obtain some
solvable conditions of 1.9 in Section 3. However, we cannot exclude the possibility that the
solution we found is trivial. To this end, we make use of Morse theory to obtain the existence
and multiplicity of nontrivial solutions to 1.9. Examples are also included to illustrate the
results obtained.
2. Index Theory for Linear Systems
To establish the index theory for 1.12, we introduce the following finite dimensional
sequence space:
E u | u u0, u1, . . . , uT , uT 1τ , u0 0 uT 1 ,
2.1
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5
where un u1 n, u2 n, . . . , ud nτ ∈ Rd for every n ∈ Z0, T 1. Define the inner product
on E as follows:
u, v T
Δun, Δvn,
∀u, v ∈ E,
2.2
∀u ∈ E,
2.3
n0
by which the norm · on E can be induced by
T
2
|Δun|
u 1/2
,
n0
where ·, · is the usual inner product on Rd , and | · | is the usual norm on Rd .
Define a linear map Γ : E → RdT by
Γu u1 1, u1 2, . . . , u1 T , u2 1, u2 2, . . . , u2 T , . . . , ud 1, ud 2, . . . , ud T τ .
2.4
It is easy to see that the map Γ defined in 2.4 is a linear homeomorphism, and E, ·, · is a
Hilbert space, which can be identified with RdT .
Define
qA u, v T
T
Δun, Δvn − Anun, vn,
n0
∀u, v ∈ E.
2.5
n1
For any u, v ∈ E, if qA u, v 0, we say that u and v are qA orthogonal. For any two subspaces
E1 and E2 of E, if qA u, v 0 for any u ∈ E1 and v ∈ E2 , we say that E1 and E2 are qA
orthogonal.
For any subspace E1 of E, we say that qA is positive definite or negative definite on
E1 if qA u, u > 0 or qA u, u < 0 for all u ∈ E1 \ {0}. And if qA u, u 0 for all u ∈ E1 , then
E1 is called a null subspace of E.
Proposition 2.1. For any A : Z1, T → GLs Rd , the following results hold.
1 There are {λi A}m
i1 ⊂ R with λ1 A < λ2 A < · · · < λm A such that
Δ2 un − 1 An λi AId un 0,
n ∈ Z1, T ,
u0 0 uT 1
2.6
has a nontrivial solution. If Ei A denotes the subspace of solutions with respect to λi A,
m
then dim Ei A : ni ≤ d and E i1 Ei A.
2 The space E has a qA orthogonal decomposition
E E A
E0 A
E− A
2.7
such that qA is positive definite, negative definite, and null on E A, E− A, and E0 A,
respectively.
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To prove Proposition 2.1, we need the following lemma.
1
∈ E, the following inequalities hold.
Lemma 2.2. For any u {un}Tn0
4 sin2
T
T
T
π
π
|un|2 ≤
|Δun|2 ≤ 4 cos2
|un|2 .
2T 1 n1
2T
1
n0
n1
2.8
Proof. Note that
T
Δun, Δun n0
T
T −1
2un, un − 2un, un 1 AΓu, Γu,
n1
2.9
n1
where
⎛
⎛
B
0 ··· 0
⎞
⎟
⎜
⎜ 0 B ··· 0 ⎟
⎟
⎜
A⎜
,
⎟
⎜· · · · · · · · · · · ·⎟
⎠
⎝
0 0 · · · B dT ×dT
2 −1 0 · · · 0
⎜
⎜−1
⎜
⎜
⎜0
⎜
B⎜
⎜· · ·
⎜
⎜
⎜0
⎝
0
2 −1 · · · 0
−1 2 · · · 0
··· ··· ··· ···
0
0 ··· 2
0
0 · · · −1
0
⎞
⎟
0⎟
⎟
⎟
0⎟
⎟
⎟ .
· · ·⎟
⎟
⎟
−1⎟
⎠
2 T ×T
2.10
Assume that λ is an eigenvalue of B and that ξ ξ1 , ξ2 , . . . , ξT τ is an eigenvector associated
1
as
to λ. Define the sequence {vn}Tn0
vi ξi ,
i 1, 2, . . . , T,
v0 0 vT 1.
2.11
1
Then {vn}Tn0
satisfies
Δ2 vn − 1 λvn 0,
n ∈ Z1, T ,
v0 0 vT 1.
2.12
Equation 2.12 has a nontrivial solution if and only if
λ λk 4 sin2
kπ
,
2T 1
k 1, 2, . . . , T ;
2.13
see 33. So σA σB {λ1 , λ2 , . . . , λT } with λ1 < λ2 < · · · < λT and
λmin min{λ1 , λ2 , . . . , λT } 4 sin2
λmax
π
λ1 ,
2T 1
π
λT .
max{λ1 , λ2 , . . . , λT } 4 cos
2T 1
2
2.14
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7
Noticing that for any real symmetric dT × dT matrix A, we have
λ1 Γu, Γu ≤ AΓu, Γu ≤ λT Γu, Γu,
Since Γu, Γu T
n1
∀u ∈ RdT .
2.15
|un|2 , the inequalities 2.8 now follow from 2.9 and 2.15.
Remark 2.3. In the following, we rewrite 2.8 as
2.16
λ1 |u|2 ≤ u2 ≤ λT |u|2
for simplicity.
Proof of Proposition 2.1. 1 We claim that the norm · λ0 induced by the inner product
u, vλ0 :
T
Δun, Δvn n0
T
λ0 Id − Anun, vn,
∀u, v ∈ E
2.17
n1
is equivalent to · , where λ0 is a positive number satisfying λ0 Id > A. In fact, it is easy to see
that there exists c ∈ 0, ∞ such that
0≤
T
T
T
c
c
λ0 Id − Anun, un ≤ c |un|2 ≤
|Δun|2 u2 .
λ1 n0
λ1
n1
n1
2.18
Hence
u2 ≤ u2λ0 ≤
1
c
u2 .
λ1
2.19
Define a bilinear function
au, v T
un, vn,
∀u, v ∈ E,
2.20
n1
and then
|au, v| ≤
T
|un|2
1/2 T
2
|vn|
n1
n1
1/2
≤
1
1
uv ≤ uλ0 vλ0 .
λ1
λ1
2.21
Hence, there exists a unique continuous linear operator Kλ0 : E → E satisfying
T
n1
un, vn u, Kλ0 vλ0 .
2.22
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It is easy to see that Kλ0 is self-adjoint, and hence all the eigenvalues of Kλ0 are real. Therefore,
there exist μi , i 1, 2, . . . , m and eij , j 1, 2, . . . , ni such that
eij , elk
λ0
⎧
⎨1,
i l and j k,
⎩0,
i
/ l or j / k,
where ni is the multiplicity of μi with
i
Kλ0 eij μi eij ,
2.23
ni dT . By 2.22 and 2.23 we have
T
μi eij , u λ0 eij n, un ,
2.24
∀u ∈ E.
n1
In particular, μi Tn1 |eij n|2 > 0. Without loss of generality we assume that μi is strictly
monotonously decreasing, that is, μ1 > μ2 > · · · > μm . Denote λi A 1/μi − λ0 and Ei A m
T 1
i
, then E span{eij }nj1
i1 Ei A. We claim that for every λi A, eij {eij n}n0 ∈ E is a
nontrivial solution of 2.6. In fact, by 2.24, for any u ∈ E, we have
μi
T
T
T
Δeij n, Δun μi
eij n, un ,
λ0 Id − Aneij n, un n0
n1
2.25
n1
and since μi > 0, the above equality means
T 1
Δ2 eij n − 1 An Id − λ0 Id eij n, un 0,
μi
n1
∀u ∈ E.
2.26
Therefore eij satisfies 2.6. Now, we have proved the first result of Proposition 2.1 except
dim Ei A ni ≤ d.
Set un y1 n, Δun − 1 −y2 n; then 2.6 is equivalent to
Δy1 n −y2 n 1,
n ∈ Z0, T ,
Δy2 n An λi AId y1 n,
2.27
n ∈ Z1, T ,
y1 0 0 y1 T 1,
which is also equivalent to
yn 1 Bnyn,
n ∈ Z1, T ,
2.28
y1 0 0 y1 T 1,
where
y1 n
yn y2 n
,
Bn Id − λi AId − An −Id
λi AId An
Id
.
2.29
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9
Since detBn ≡ 1, Bn is a nonsingular 2d × 2d matrix for every n. So, we can assume that
Φn is the fundamental matrix of equation yn 1 Bnyn satisfying Φ0 I2d . The
c1
general solution of yn 1 Bnyn can be given by yn Φnc, where c c ∈ R2d
2
and ci ∈ Rd , i 1, 2. Set
Φn Φ11 n Φ12 n
Φ21 n Φ22 n
,
where Φij n is d × d matrix,
i, j 1, 2,
2.30
then y1 n Φ11 nc1 Φ12 nc2 . By y1 0 0 y1 T 1 and Φ0 I2d , we have c1 0 and
Ei A ∼
c2 ∈ Rd | Φ12 T 1c2 0 ⊆ Rd .
2.31
Hence, dim Ei A ni ≤ d.
2 For any u ∈ E with u i,j cij eij , by 2.23 and 2.24, we have
qA u, u u, uλ0 − λ0 u, Kλ0 uλ0 2
2
cij − λ0 μi cij 2 λi Aμi cij .
i,j
i,j
2.32
Hence, if we denote
E A u cij eij | cij 0, if λi A ≤ 0 ,
cij eij | cij 0, if λi A E0 A u /0 ,
2.33
cij eij | cij 0, if λi A ≥ 0 ,
E− A u then the results hold.
Definition 2.4. For any A : Z1, T → GLs Rd , define the index of A as iA : dim E− A,
and define the nullity of A as νA : dim E0 A.
In the following we shall discuss the properties of iA, vA.
Proposition 2.5. For any A : Z1, T → GLs Rd , one has the following.
1 νA is the dimension of the solution subspace of 1.12, and νA ∈ Z0, d.
2 iA λi A<0 ni , where λi and ni are defined in the proof of Proposition 2.1.
Proof. 1 By Proposition 2.5, if λi A /
0 for any i, then 1.12 has only a trivial solution, and
hence E0 A {0}, νA 0. If λi A 0 for some i with multiplicity ni , then by the proof of
Proposition 2.1, Ei A is the solution subspace of 1.12 and νA : dim Ei A ∈ Z1, d.
2 By the proof of Proposition 2.1, E− A λi A<0 Ei A, and Ei A and Ej A are
j. Hence the result holds.
qA orthogonal if i /
Remark 2.6. By 1 of Proposition 2.5, νA ≥ 0, and νA 0 if and only if λi A /
0 for any
i ∈ Z1, dT which holds if and only if 1.12 has only a trivial solution.
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Discrete Dynamics in Nature and Society
Proposition 2.7. For any A1 , A2 : Z1, T → GLs Rd , the following results hold.
1 If A1 ≤ A2 , then iA1 ≤ iA2 .
2 If A1 ≤ A2 , then iA1 νA1 ≤ iA2 νA2 .
3 If A1 < A2 , then iA1 νA1 ≤ iA2 .
To prove Proposition 2.7, we firstly prove the following lemma.
Lemma 2.8. Let E1 be a subspace of E satisfying
qA u, u ≤ 0,
∀u ∈ E1 ,
2.34
and then
dim E1 ≤ iA νA.
2.35
Moreover, if
qA u, u < 0,
∀u ∈ E1 \ {0},
2.36
then
dim E1 ≤ iA.
2.37
Proof. Without loss of generality we can assume that dim E1 k ≥ 1 and E1 span{e1 , e2 , . . . , ek }. Let ei ei ei∗ , where ei∗ ∈ E− A ⊕ E0 A, ei ∈ E A. To prove
iA νA ≥ k, we only need to prove that e1∗ , e2∗ , . . . , ek∗ is linear independent. If not, there
exist not all zero constants α1 , α2 , . . . , αk such that ki1 αi ei∗ 0. So e : ki1 αi ei ki1 αi ei ∈
E A, and hence qA e, e > 0. This is a contradiction to 2.34. This implies that e1∗ , e2∗ , . . . , ek∗
is linear independent and iA νA : dim E− A dim E0 A dimE− A ⊕ E0 A ≥ k.
The first part is proved.
To prove the second part, let ei ei ei− ei0 , where ei∗ ∈ E∗ A, ∗ , 0, −. To prove
iA ≥ k, we only need to prove that e1− , e2− , . . . , ek− is linear independent. If not, there exist not
all zero constants c1 , c2 , . . . , ck such that ki1 ci ei− 0. So, e : ki1 ci ei ki1 ci ei ci ei0 ∈
E A ⊕ E0 A and qA e, e ≥ 0. This is a contradiction to 2.36.
Proof of Proposition 2.7. For any u ∈ E, denote u u u0 u− , where u∗ ∈ E∗ A1 , ∗ , −, 0.
1 From Lemma 2.8, we only need to show that
qA2 u, u < 0,
∀u ∈ E− A1 \ {0}.
2.38
0, then
In fact, for every u u− ∈ E− A1 , if u− /
qA2 u, u ≤ qA1 u, u qA1 u− , u− < 0.
2.39
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11
2 From Lemma 2.8, we only need to show that
qA2 u, u ≤ 0,
∀u ∈ E− A1 ⊕ E0 A1 .
2.40
In fact, for every u u− u0 ∈ E− A1 ⊕ E0 A1 , one has
qA2 u, u ≤ qA1 u, u qA1 u0 , u0 qA1 u− , u− ≤ 0.
2.41
3 From Lemma 2.8, we only need to show that
qA2 u, u < 0,
∀u ∈ E− A1 ⊕ E0 A1 \ {0}.
2.42
0, then
In fact, for every u u− u0 with u− ∈ E− A1 , u0 ∈ E0 A1 , if u− /
qA2 u, u ≤ qA1 u, u qA1 u0 , u0 qA1 u− , u− < 0.
2.43
T 1
If u− 0, u0 / 0, then {u0 n}n0 is a nontrivial solution of
Δ2 un − 1 A1 nun 0,
n ∈ Z1, T ,
u0 0 uT 1.
2.44
From A1 < A2 we have
qA2 u0 , u0 < qA1 u0 , u0 0.
2.45
Hence 2.42 holds.
Proposition 2.9. If U ∈ Od, that is, U ∈ GLs Rd and Uτ U Id , then for any A : Z1, T →
GLs Rd , iUτ AU iA, νUτ AU νA. In particular, for any A ∈ GLs Rd , we have
iA d
{k ∈ Z1, T | λk < ai },
i1
νA d
{k ∈ Z1, T | λk ai },
2.46
i1
where λk is given by 2.13, and {ai }di1 σA is the set of eigenvalues of A.
Proof. Firstly, we claim that
λi Uτ AU λi A.
2.47
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In fact, since λ0 Id > A if and only if λ0 Id > Uτ AU, we can choose λ0 Uτ AU λ0 A. By
2.22 and 2.23, it is easy to see that μi Uτ AU μi A, and hence 2.47 holds. Therefore,
by Definition 2.4,
iUτ AU iA,
νUτ AU νA.
2.48
Since
E∗ diag{A1 , A2 } ∼
E∗ A1 ⊕ E∗ A2 ,
∗ , 0, −,
2.49
we have
i diag{A1 , A2 } iA1 iA2 ,
ν diag{A1 , A2 } νA1 νA2 .
2.50
Note that the scalar eigenvalue problem
Δ2 yn − 1 λyn 0,
n ∈ Z1, T ,
y0 0 yT 1
2.51
has a nontrivial solution if and only if λ λk 4 sin2 kπ/2T 1, k ∈ Z1, T. By
Proposition 2.1 and Definition 2.4, we see that for any α ∈ R,
iα {k ∈ Z1, T | λk < α},
να {k ∈ Z1, T | λk α}.
2.52
Since {ai }di1 is the set of eigenvalues of A, there exists an orthogonal matrix U such that
Uτ AU diag{a1 , a2 , . . . , ad }.
2.53
From 2.48, 2.50, and 2.52 we have
iA d
{k ∈ Z1, T | λk < ai },
i1
νA d
{k ∈ Z1, T | λk ai }.
2.54
i1
This completes the proof.
Proposition 2.10. For any A : Z1, T → GLs Rd with iA 0, one has
T
T
Anun, un,
|Δun|2 ≥
n0
n1
And the equality holds if and only if u ∈ E0 A.
∀u ∈ E.
2.55
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Proof. For any u ∈ E with u i,j
13
cij eij , we have
T
T
2
Anun, un λi Aμi cij .
|Δun|2 n0
2.56
i,j
n1
Because iA 0, by definition, λi A ≥ 0 for any i. So the inequality holds. And the equality
0, that is, u ∈ E0 A.
holds if and only if cij 0 as λi A /
By now, we have proved the monotonicity and have offered the computation formulae
of the indices. These will play an important role in discussing nonlinear Hamiltonian systems
in the next section. In the end of this section, we shall introduce the relative Morse index,
which is a precise expression of the number iA2 − iA1 as A2 > A1 .
Definition 2.11. For any A1 , A2 : Z1, T → GLs Rd with A1 < A2 , define
IA1 , A2 νA1 λA2 − A1 .
λ∈0,1
2.57
If A1 α1 Id , A2 α2 Id , where α1 < α2 are two real numbers, then by Proposition 2.9,
we have
Iα1 Id , α2 Id να1 Id λα2 − α1 Id d{k ∈ Z1, T | λk ∈ α1 , α2 },
λ∈0,1
iα1 Id d{k ∈ Z1, T | λk < α1 },
2.58
iα2 Id d{k ∈ Z1, T | λk < α2 }.
So
Iα1 Id , α2 Id iα2 Id − iα1 Id .
2.59
This gives us a steer toward the following result.
Proposition 2.12. For any A1 , A2 : Z1, T → GLs Rd with A1 < A2 , one has
IA1 , A2 iA2 − iA1 .
2.60
Proof. Denote iλ : iA1 λA2 − A1 for λ ∈ 0, 1, νλ : νA1 λA2 − A1 ; then to prove
2.60, we only need to prove that
iλ 0 iλ νλ,
iλ − 0 iλ,
∀λ ∈ 0, 1,
∀λ ∈ 0, 1
2.61
2.62
hold. In fact, if 2.61 and 2.62 hold, then the function λ → iλ is integer-valued, left
continuous, and nondecreasing. So, for any λ1 ∈ 0, 1, iλ1 − i0 must equal to the sum
14
Discrete Dynamics in Nature and Society
of the jumps iλ incurred in 0, λ1 . By 2.61 and 2.62, this is precisely the sum of νλ,
0 ≤ λ < λ1 , that is,
iλ1 − i0 νλ.
λ∈0,λ1 2.63
Hence, if we choose λ1 1, then 2.60 holds.
From 3 of Proposition 2.7, to prove 2.61, we only need to prove iλ 0 ≤ iλ νλ
which is also equivalent to dT − iλ − νλ ≤ dT − iλ 0. For any s ∈ 0, 1, set m s dT − is − νs, we only need to prove
m λ ≤ m λ 0 νλ 0.
2.64
Similar to the proof of Lemma 2.8, it is easy to know that for ε > 0 sufficiently small, if
qBλε u, u ≥ 0,
∀u ∈ E λ,
2.65
then 2.64 holds, where E λ E A1 λA2 −A1 , Bλε n A1 nλεA2 n−A1 n.
While as ε > 0 is sufficiently small and u ∈ E λ, we have
qBλε u, u qBλ u, u − ε
T
A2 n − A1 nun, un
n1
T
qBλ u , u − ε A2 n − A1 nun, un ≥ 0,
2.66
n1
where Bλ n A1 n λA2 n − A1 n. Hence iλ 0 ≤ iλ νλ.
On the other hand, from 1 of Proposition 2.7, to prove 2.62, we only need to prove
iλ ≤ iλ − 0. By Lemma 2.8, to prove iλ ≤ iλ − 0, we only need to prove
qBλ−ε u, u < 0,
∀u ∈ E− λ \ {0},
2.67
where ε > 0 is sufficiently small, Bλ−ε n A1 n λ − εA2 n − A1 n. And as ε > 0 is
sufficiently small, we have
qBλ−ε u, u qBλ u, u ε
T
A2 n − A1 nun, un
n1
qBλ
T
u− , u− ε A2 n − A1 nun, un < 0,
n1
where Bλ n A1 n λA2 n − A1 n. This completes the proof.
2.68
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15
Proposition 2.13. For any A : Z1, T → GLs Rd , there exists ε0 > 0 such that for any ε ∈ 0, ε0 ,
one has
νA εId 0,
2.69
νA − εId 0,
2.70
iA − εId iA,
2.71
iA εId iA νA.
2.72
Proof. From Proposition 2.12 we have iA Id iA IA, A Id . From Definition 2.11 and
Proposition 2.12, we know that IA, A Id λ∈0,1 νA λId is finite, and then there must
exist some ε0 > 0 such that for any ε ∈ 0, ε0 , νA εId 0 and
iA εId iA νA λεId iA νA.
λ∈0,1
2.73
This proves 2.69 and 2.72.
To prove 2.70 and 2.71, note that IA − Id , A iA − iA − Id and
IA − Id , A νA − 1 − λId λ∈0,1
νA − λId .
λ∈0,1
2.74
Since IA − Id , A is finite, there exists ε0 > 0 such that for any ε ∈ 0, ε0 , νA − εId 0. Hence
iA − εId iA − IA − εId , A iA −
νA − λεId iA.
λ∈0,1
2.75
This proves 2.70 and 2.71.
3. Main Results
In this section, firstly, we shall obtain the existence of solutions to 1.9 by using both the
index theory in Section 2 and Leray-Schauder principle. Then, we obtain the multiplicity of
solutions to 1.9 by using Morse theory.
Theorem 3.1. Assume that
1 there exist A : Z1, T × Rd → GLs Rd and h : Z1, T × Rd → Rd which are both
continuous with respect to the second variable, where hn, z o|z| as |z| → ∞ for
every n ∈ Z1, T and
∇V n, z An, zz hn, z;
3.1
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Discrete Dynamics in Nature and Society
2 there exist A1 , A2 : Z1, T → GLs Rd satisfying A1 ≤ A2 , iA1 iA2 , νA2 0
such that
A1 n ≤ An, z ≤ A2 n,
3.2
∀z ∈ Rd
for every n ∈ Z1, T . Then 1.9 has at least one solution.
To prove Theorem 3.1, we need the following Leray-Schauder principle; see 34 for
detailed proof.
Lemma 3.2. Assume that X, ·X is a Banach space and that Φ : X → X is completely continuous.
If the set {xX | x ∈ X, x λΦx, 0 < λ < 1} is bounded, then Φ must have a fixed point in a closed
ball BR in X, where
BR {x | x ∈ X, xX ≤ R},
3.3
R sup{xX | x λΦx, 0 < λ < 1}.
Proof of Theorem 3.1. Assume that 3.2 holds. Since νA2 0, from 1 of Proposition 2.5, we
know that the system
Δ2 un − 1 A2 nun 0,
n ∈ Z1, T ,
u0 0 uT 1
3.4
has only a trivial solution. Define Γ1 : E → E as
3.5
Γ1 un Δ2 un − 1 A2 nun;
then Γ1 is an invertible operator. Define Γ2 : E → E as
Γ2 un A2 nun − ∇V n, un;
3.6
then finding the solutions to 1.9 is equivalent to finding solutions to
Γ1 u − Γ2 u 0
3.7
in E, which is also equivalent to finding the fixed points of Γ−1
1 Γ2 in E since Γ1 is invertible. By
Lemma 3.2, we only need to prove that the possible solutions to
Δ2 un − 1 1 − λA2 nun λ∇V n, un 0,
u0 0 uT 1
n ∈ Z1, T ,
3.8
Discrete Dynamics in Nature and Society
17
are priori bounded with respect to the norm · in E, where λ ∈ 0, 1. If not, there exist
{uk } ⊂ E, {λk } ⊂ 0, 1 with uk → ∞ such that
Δ2 uk n − 1 1 − λk A2 nuk n λk ∇V n, uk n 0,
n ∈ Z1, T ,
uk 0 0 uk T 1.
3.9
Denote vk n uk n/uk , Bk n 1 − λk A2 n λk An, uk n, ek n λk ∇V n, uk n −
An, uk nuk n/uk ; then 3.9 is equivalent to
Δ2 vk n − 1 Bk nvk n ek n 0,
n ∈ Z1, T ,
3.10
vk 0 0 vk T 1.
From 3.1, ek → 0 as uk → ∞. We may assume that vk → v0 , λk → λ0 , and bijk → bij ,
k
where Bk n bij d×d n. Denote B0 n bij nd×d ; let k → ∞ in 3.10; we have
Δ2 v0 n − 1 B0 nv0 n 0,
n ∈ Z1, T ,
3.11
v0 0 0 v0 T 1.
On the other hand, 3.2 implies that A1 ≤ Bk ≤ A2 , and hence A1 ≤ B0 ≤ A2 . By iA2 iA1 ,
νA2 0, and Proposition 2.7, we have νA1 νA2 νB0 0. This contradicts the fact
that 3.11 has a nontrivial solution.
Example 3.3. Let
d
d
1
V n, z Fi zi aii nz2i aij nzi zj 2 i1
i1
1≤i<j≤d
d
3/4
z2i
1
sin n,
3.12
i1
t
where z z1 , z2 , . . . , zd τ , Fi t 0 sfi sds, fi : R → 0, α is continuous and α > 0,
aij n aji n, i 1, 2, . . . , d, j 1, 2, . . . , d. Set
An aij n d×d Uτ diag{α1 n, α2 n, . . . , αd n}U,
3.13
where U ∈ Od. Since
∇V n, z z1 f1 z1 , z2 f2 z2 , . . . , zd fd zd τ
3
Anz 2
3
diag f1 z1 , f2 z2 , . . . , fd zd z Anz 2
d
−1/4
z2i
1
z sin n
i1
d
i1
3.14
−1/4
z2i 1
z sin n,
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Discrete Dynamics in Nature and Society
V n, z satisfies 3.1 with
An, z An diag f1 z1 , f2 z2 , . . . , fd zd ,
A1 n An,
hn, z 3
2
A2 n An αId ,
d
3.15
−1/4
z2i 1
z sin n.
i1
If ναi 0 for every i ∈ Z1, d, then νA1 0. By Proposition 2.13, if α > 0 is small enough,
then νA2 0 and iA1 iA2 . Hence, by Theorem 3.1, 1.9 has at least one solution. In
particular, if we choose fi t αsin t2i and αi n ≡ αi ∈ R, αi , αi α ∩ {λk }Tk1 ∅ for every
i 1, 2, . . . , d, then iA1 iA2 , νA1 νA2 0. And hence 1.9 has at least one solution.
Theorem 3.4. In assumption 3.1 if An, z An satisfying νA / 0 and
z, hn, z ≥ c1 |z|α − b1 ,
3.16
|hn, z| ≤ c2 |z|α−1 b2 ,
where c1 , c2 , b1 , b2 are all positive constants and 1 ≤ α < 2, then 1.9 has at least one solution.
Proof. From Proposition 2.13, there exists ε > 0 such that iA εId iA νA and νA εId 0 for any A : Z1, T → GLs Rd . Denote A1 A εId , by Lemma 3.2, we only need
to prove that the possible solutions to
Δ2 un − 1 λA1 nun 1 − λAnun 1 − λhn, un 0,
n ∈ Z1, T ,
u0 0 uT 1
3.17
are priori bounded with respect to the norm · in E. If not, there exist {uk } ⊂ E, λk ∈ 0, 1
with uk → ∞ such that
Δ2 uk n − 1 λk A1 nuk n 1 − λk Anuk n 1 − λk hn, uk n 0,
n ∈ Z1, T ,
uk 0 0 uk T 1.
3.18
Denote vk n uk n/uk , we may assume that vk → v0 and λk → λ0 . Hence, v v0 is a
nontrivial solution to
Δ2 v0 n − 1 λ0 A1 nv0 n 1 − λ0 Anv0 n 0,
v0 0 0 v0 T 1
n ∈ Z1, T ,
3.19
which implies that νA ελ0 Id /
0. We claim that λ0 0. If not, λ0 ∈ 0, 1, then A < A λ0 εId ≤ A εId . From Proposition 2.7, we have iA εId iA νA ≤ iA λ0 εId and
Discrete Dynamics in Nature and Society
19
iA εId iA λ0 εId . However, from Proposition 2.7, we also have iA λ0 εId νA λ0 εId ≤ iA εId νA εId which implies that νA λ0 εId 0, a contradiction! Hence
λ0 0 and
Δ2 v0 n − 1 Anv0 n 0,
n ∈ Z1, T ,
3.20
v0 0 0 v0 T 1.
From 3.18, we have
T Δ2 uk n − 1 Anuk n, v0 n
n1
T
ελk uk n, v0 n n1
T
3.21
1 − λk hn, uk n, v0 n 0.
n1
Therefore, from 3.16, 3.20, and 3.21, for k large enough,
0≥
T
hn, uk n, v0 n
n1
T T
uk n
hn, uk n, v0 n − vk n
hn, uk n,
uk n1
n1
≥ uk −1
3.22
T c2 |uk n|α−1 b2 .
c1 |uk n|α − b1 − v0 − vk ∞
T
n1
n1
Dividing uk α−1 at both sides, we have
0≥
T T
c2 |vk n|α−1 b2 uk 1−α −→ c1 |v0 n|α .
c1 |vk n|α − b1 uk −α − v0 − vk ∞
T
n1
n1
n1
3.23
This is a contradiction since v0 /
0 and c1 > 0. The proof is complete.
If ∇V n, 0 ≡ 0, then u ≡ 0 is a solution to 1.9. As usual we call this solution the
trivial solution. It is much regretted that we do not know if the solution we found is not the
trivial one in Theorems 3.1 and 3.4. In the following, we will obtain the existence of nontrivial
solutions to 1.9 by using Morse theory.
Theorem 3.5. Assume the following
1 V : Z1, T × Rd → R is C2 with respect to the second variable, and
∇V n, z A1 nz o|z|,
for every n ∈ Z1, T with νA1 0.
as |z| −→ ∞
3.24
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Discrete Dynamics in Nature and Society
∈ ZiA0 , iA0 νA0 . Then 1.9 has at
2 ∇V n, 0 ≡ 0, A0 n V n, 0, and iA1 /
least one nontrivial solution.
3 Moreover, if we further assume that νA0 0, |iA1 − iA0 | ≥ d, then 1.9 has at least
two nontrivial solutions.
To prove Theorem 3.5, we need some results on Morse theory. Let E be a real Hilbert
space and f ∈ C1 E, R. As in 2, denote
f c u ∈ E | fu ≤ c ,
Kc u ∈ E | f u 0, fu c
3.25
for c ∈ R. The following is the definition of the Palais-Smale condition the PS condition for
short.
Definition 3.6. The functional f satisfies the PS condition if any sequence {um } ⊂ E such that
{fum } is bounded and f um → 0 as m → ∞ has a convergent subsequence.
Let u0 be an isolated critical point of f with fu0 c ∈ R, and let U be a neighborhood
of u0 ; the group
Cq f, u0 : Hq f c ∩ U, f c ∩ U \ {u0 } ,
q 1, 2, . . .
3.26
is called the qth critical group of f at u0 , where Hq A, B denotes qth singular relative
homology group of the pair A, B over a field F, which is defined to be quotient Hq A, B Zq A, B/Bq A, B, where Zq A, B is the qth singular relative closed chain group and
Bq A, B is the qth singular relative boundary chain group.
For any two regular values a < b, if K ∩ f −1 a, b {u1 , u2 , . . . , ul }, denote Mq l
b
a
i1 dim Cq f, ui and βq dim Hq f , f . The following results play an important role in
proving Theorem 3.5; see 2 for the detailed proof.
Lemma 3.7. Assume that f ∈ C2 E, R satisfies the (PS) condition. Then one has the following Morse
inequalities:
Mq − Mq−1 · · · −1q M0 ≥ βq − βq−1 · · · −1q β0
3.27
for q 0, 1, 2, . . . . One also has the following Morse equality:
∞
q0
Mq tq ∞
βq tq 1 tQt,
q0
where Qt is a polynomial with nonnegative integer coefficients.
3.28
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21
Lemma 3.8. Assume that f ∈ C2 E, R and that u0 is an isolated critical point of f with finite Morse
index μu0 and nullity νu0 . Thenone has the following.
1 For any q /
∈ Zμu0 , μu0 νu0 , Cq f, u0 ∼
0.
2 If u0 is a nondegenerate critical point, then
Cq f, u0 ∼
δq,μu0 F,
q 0, 1, 2, . . . .
3.29
3 If f has a minimal value at u0 , then
Cq f, x0 ∼
δq,0 F,
q 0, 1, 2, . . . .
3.30
4 If there exist integers q1 /
q2 such that Cq1 f, u0 0 and Cq2 f, u0 0, then |q1 − q2 | ≤
νu0 − 2.
Now, Define
T
T
1
|Δun|2 − V n, un,
2 n0
n1
fu u ∈ E.
3.31
Then the functional f is C2 with
T
T
f u, v Δun, Δvn − ∇V n, un, vn
n0
n1
T −
Δ2 un − 1 ∇V n, un, vn ,
3.32
∀u, v ∈ E.
n1
So solutions to 1.9 are precisely the critical points of f.
Lemma 3.9. Under assumptions of Theorem 3.5, there exist R0 > 0, R1 > 0 and f satisfying the
following conditions.
1 f u 0 implies u ≤ R0 .
2 f and f have the same critical set.
3 If u ≥ R1 , then fu
1/2Lu, u.
Proof. Define L and g on E as
Lu, v T
T
Δun, Δvn − A1 nun, vn,
n0
n1
1
gu fu − Lu, u.
2
3.33
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Then
f u, v Lu, v g u, v .
3.34
−1
Assumption νA1 0 implies that L is invertible. Taking ε1 L−1 /2, there exists R0 > 0
such that if u > R0 , then
g u < ε1 u.
3.35
So, as u > R0 , we have
f u ≥ L−1
−1
u − g u > ε1 u,
3.36
that is, no critical point is outside the ball BR0 .
To prove 2 and 3, let ρ ∈ C∞ R, 0, 1 satisfy
ρt ⎧
⎨1,
t ≤ 0,
⎩0,
t≥1
3.37
with 0 ≤ ρt ≤ 1 and max |ρ t| ≤ 3/2. Let ε L−1 −1 /5, then there exists c1 ≥ 0 such that
g u < εu c1 .
3.38
Hence,
gu ≤
!1
g su, u ds g0
0
≤
! 1
εsu2 c1 u ds g0
3.39
0
ε
u2 c1 u g0.
2
Therefore, there exists c2 > 0 such that
gu ≤ εu2 c2 ,
∀u ∈ E.
3.40
Define
#
"
c1 3/2
,
R > max 1, R0 ,
3.41
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23
where R0 is defined above and
λ max{c2 , 1} · R.
3.42
If R ≤ u ≤ λ R, then
c2
< 1,
λ
u
≤ 2.
λ
3.43
t−R
,
λ
3.44
Let
ρλ t ρ
and define
1
Lu, u ρλ ugu.
fu
2
3.45
Then the function fu
satisfies 2 and 3. In fact,
fu
⎧
⎪
⎨fu,
u ≤ R,
⎪
⎩ 1 Lu, u,
2
u ≥ λ R.
3.46
0 as R ≤ u ≤ λ R. However,
The only thing we have to check is that f u /
1 u − R
u
u − R gu
g u
f u Lu ρ
ρ
λ
λ
λ
u
−1
3 εu2 c2 − εu c1 ≥ L−1 u −
2λ
3ε
3c2
5εu − εu −
u2 − c1 2λ
2λ
3
≥ εu − c1 > 0.
2
3.47
Hence, let R1 λ R; the proof is completed.
From Lemma 3.9, f u 0 if and only if f u 0. Thus, in order to find solutions
Moreover, f satisfies the PS condition by
to 1.9, it suffices to find the critical points of f.
Lemma 3.9.
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Lemma 3.10. Under the assumptions of Theorem 3.5, there exist a, b with a < b such that the critical
points of f belong to f−1 a, b : {u | a < fu
< b} and
Hq fb , fa ∼
δq,iA1 F.
3.48
Proof. Define
a < min f − 1,
B2R1
b > max f 1,
f1 u B2R1
1
Lu, u,
2
3.49
where a, b are finite. Noticing that all critical points of f lie in BR0 , if u is a critical point of f,
then
≤ max f < b.
a < min f ≤ fu
BR0
BR0
3.50
By the properties of the
This implies that {u | a < fu
< b} contains all critical points of f.
b
raltive singular homology group, we have Hq fb , fa ∼
Hq f1 , f1a . However, νA1 0
implies that f1 has only critical point 0 with Morse index iA1 . From Lemma 3.8 the
conclusion holds.
Proof of Theorem 3.5. 1 By Lemma 3.10 and the Morse inequalities, f must have a critical
∈ ZiA0 , iA0 νA0 , then from Lemma 3.8, we
point u with CiA1 f, u 0. Since iA1 /
have
CiA1 f, 0 ∼
0.
3.51
And hence u /
0 is a critical point of f; that is, 1.9 has at least one nontrivial solution.
2 Since νA0 0, we have
Cq f, 0 ∼
δq,iA0 F,
q 0, 1, 2, . . . .
3.52
Now we assume that |iA1 − iA0 | ≥ d and that u is the only nonzero critical point of f. Then
from Morse equality, we have
tiA0 ∞
q0
dim Cq f, u tq tiA1 1 tQt.
3.53
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25
We necessarily have dim CiA1 f, u ≥ 1, and
either dim CiA0 −1 f, u ≥ 1,
or dim CiA0 1 f, u ≥ 1.
3.54
i If dim CiA0 −1 f, u ≥ 1, then by assumption we have iA0 − 1 / iA1 . Since the
nullity of u is less or equal to d, from Lemma 3.8, we have
|iA0 − 1 − iA1 | ≤ d − 2
3.55
which implies that |iA0 − iA1 | ≤ d − 1. This is impossible since |iA0 − iA1 | ≥ d.
ii If dim CiA0 1 f, u ≥ 1, then similar to the above proof we have |iA0 − iA1 | ≤
d − 1, also a contradiction.
Therefore, f has at least two nonzero critical points and hence 1.9 has at least two
nontrivial solutions.
Acknowledgments
The author would like to express her thanks to the referees for helpful suggestions. This
research is supported by Guangdong College Yumiao Project 2009.
References
1 J. K. Hale, Ordinary Differential Equations, vol. 34 of Pure and Applied Mathematics, Wiley-Interscience,
New York, NY, USA, 1969.
2 J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical
Sciences, Springer, New York, NY, USA, 1989.
3 W. Walter, Ordinary Differential Equations, vol. 182 of Graduate Texts in Mathematics, Springer, New
York, NY, USA, 1998.
4 R. P. Agarwal, “On multiple boundary value problems for discrete equations,” Journal of Mathematical
Analysis and Applications, vol. 96, pp. 520–534, 1983.
5 A. Lasota, “A discrete boundary value problem,” Annales Polonici Mathematici, vol. 20, pp. 183–190,
1968.
6 W. G. Kelley and A. C. Peterson, Difference Equations: An Introduction with Applications, Academic
Press, Boston, Mass, USA, 1991.
7 A. C. Peterson, “Existence and uniqueness theorems for nonlinear difference equations,” Journal of
Mathematical Analysis and Applications, vol. 125, no. 1, pp. 185–191, 1987.
8 R. P. Agarwal and D. O’Regan, “Boundary value problems for discrete equations,” Applied
Mathematics Letters, vol. 10, no. 4, pp. 83–89, 1997.
9 J. Henderson and H. B. Thompson, “Existence of multiple solutions for second-order discrete
boundary value problems,” Computers & Mathematics with Applications, vol. 43, no. 10-11, pp. 1239–
1248, 2002.
10 Z. Wan, Y. Chen, and S. S. Cheng, “Monotone methods for a discrete boundary problem,” Computers
& Mathematics with Applications, vol. 32, no. 12, pp. 41–49, 1996.
11 F. M. Atici and G. S. Guseinov, “Positive periodic solutions for nonlinear difference equations with
periodic coefficients,” Journal of Mathematical Analysis and Applications, vol. 232, no. 1, pp. 166–182,
1999.
12 R. P. Agarwal and D. O’Regan, “Nonpositone discrete boundary value problems,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 39, no. 2, pp. 207–215, 2000.
26
Discrete Dynamics in Nature and Society
13 F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic
boundary value problems,” Computers & Mathematics with Applications, vol. 45, no. 6-9, pp. 1417–1427,
2003.
14 R. P. Agarwal and D. O’Regan, “A fixed-point approach for nonlinear discrete boundary value
problems,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 115–121, 1998.
15 H. B. Thompson and C. Tisdell, “Systems of difference equations associated with boundary value
problems for second order systems of ordinary differential equations,” Journal of Mathematical Analysis
and Applications, vol. 248, no. 2, pp. 333–347, 2000.
16 T. He, W. Yang, and F. Yang, “Sign-changing solutions for discrete second-order three-point boundary
value problems,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 705387, 14 pages, 2010.
17 K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear
Differential Equations and Their Applications, 6, Birkhäuser, Boston, Mass, USA, 1993.
18 H.-H. Bin, J.-S. Yu, and Z.-M. Guo, “Nontrivial periodic solutions for asymptotically linear resonant
difference problem,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 477–488, 2006.
19 Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear
difference equations,” Science in China. Series A, vol. 46, no. 4, pp. 506–515, 2003.
20 Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order
difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419–430, 2003.
21 Z. Guo and J. Yu, “Multiplicity results for periodic solutions to second-order difference equations,”
Journal of Dynamics and Differential Equations, vol. 18, no. 4, pp. 943–960, 2006.
22 J.-S. Yu and B. Zheng, “Multiplicity of periodic solutions for second-order discrete Hamiltonian
systems with a small forcing term,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no.
9, pp. 3016–3029, 2008.
23 J. Yu, H. Bin, and Z. Guo, “Periodic solutions for discrete convex Hamiltonian systems via Clarke
duality,” Discrete and Continuous Dynamical Systems. Series A, vol. 15, no. 3, pp. 939–950, 2006.
24 Z. Zhou, J. Yu, and Z. Guo, “Periodic solutions of higher-dimensional discrete systems,” Proceedings
of the Royal Society of Edinburgh. Section A, vol. 134, no. 5, pp. 1013–1022, 2004.
25 Z. Zhou, J. Yu, and Z. Guo, “The existence of periodic and subharmonic solutions to subquadratic
discrete Hamiltonian systems,” The ANZIAM Journal, vol. 47, no. 1, pp. 89–102, 2005.
26 R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular and nonsingular
discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol.
58, no. 1-2, pp. 69–73, 2004.
27 X. Cai and J. Yu, “Existence theorems for second-order discrete boundary value problems,” Journal of
Mathematical Analysis and Applications, vol. 320, no. 2, pp. 649–661, 2006.
28 L. Jiang and Z. Zhou, “Existence of nontrivial solutions for discrete nonlinear two point boundary
value problems,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 318–329, 2006.
29 H. Liang and P. Weng, “Existence and multiple solutions for a second-order difference boundary
value problem via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 326, no.
1, pp. 511–520, 2007.
30 J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,”
Journal of Differential Equations, vol. 231, no. 1, pp. 18–31, 2006.
31 J. Yu and Z. Guo, “Boundary value problems of discrete generalized Emden-Fowler equation,” Science
in China. Series A, vol. 49, no. 10, pp. 1303–1314, 2006.
32 B. Zheng and Q. Zhang, “Existence and multiplicity of solutions of second-order difference boundary
value problems,” Acta Applicandae Mathematicae, vol. 110, no. 1, pp. 131–152, 2010.
33 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd
edition, 2000.
34 D. J. Guo, Nonlinear Functional Analysis, Shandong Publishing House of Science and Technology, Jinan,
China, 2001.